"Heshikiri" could be translated as "pressure slash," while Hasebe is the name of the swordsmith who eventually founded the Hasebe school.
The legend goes that Nobunaga was insulted by Kannai, his tea ceremony page, and was so enraged that he tried to execute him. Kannai tried to hide in a cupboard, but Nobunaga cut through the wood and killed him simply by letting the weight of the blade fall down.

Obtained

へし切長谷部、と言います。主命とあらば、何でもこなしますよ。

My name is Heshikiri Hasebe. If it's an order from the Master, I will accomplish anything you wish.

Hasebe might mean to ask his Master not to forget him after sending out another sword to get stronger or he could be hoping the sword that's being sent out won't forget him. It's pretty vague, but I think the first case is more likely.

To Master,
How are you?
Thank you very much for giving me your approval to go on this training journey.
Now that it's come to this, as Master's katana I intend to improve my skills more than anyone and come back without bringing you shame.
However, I do not think I have been led here only for training....
... Why is that, I wonder. It's as if destiny itself has ended up bringing me to Azuchi.

As I ended up specifically being sent to Azuchi, confronting that man1 is a situation that I cannot avoid.
Or rather, is it possible that I can only find my true strength by overcoming that man?
The problem is, if I lose my composure, won't I end up attacking him?

.... It's just a joke. I am your katana. Therefore, I am a sword whose purpose is to protect history.

Now that I've confronted that man, I feel... somewhat let down. The reason I was gifted to the Kuroda family. Even though I have always thought about it.

....That man, he was cautious of Josui-sama's2 quick wit. I was given as an offering of good will. Of course, if that is so, I was definitely a rather important necessity.

If it's cleared up, I don't have to worry about it anymore. From now on, you are the only master I live for.

↑ 2: "Josui" is the Buddhist priest name of his master Kuroda Yoshitaka. In 1587, Christianity was banned from being practiced. As a result, Yoshitaka was forced to renounce Christianity and joined the Buddhist priesthood.